A one-year course in probability theory and the theory of random processes, taught at Princeton University to undergraduate and graduate students, forms the core of the content of this book. It is structured in two parts: the first part providing a detailed discussion of Lebesgue integration, Markov chains, random walks, laws of large numbers, limit theorems, and their relation to Renormalization Group theory. The second part includes the theory of stationary random processes, martingales, generalized random processes, Brownian motion, stochastic integrals, and stochastic differential equations. One section is devoted to the theory of Gibbs random fields. This material is essential to many undergraduate and graduate courses. The book can also serve as a reference for scientists using modern probability theory in their research.
Part I Probability Theory: Spaces of Elementary Outcomes. Random Variables and their Distributions.- Sequences of Independent Trials. Lebesgue Integral and Mathematical Expectation. Conditional Probabilities and Independence.- Markov Chains.- Random Walks on the Lattice Zd.- Laws of Large Numbers.- Weak Convergence of Measures.- Characteristic Functions.- Limit Theorems.- Several Interesting Problems. Part II Random Processes: Basic Concepts.- Conditional Expectations and martingales.- Random Processes which are Stationary in the Wide Sense.- Random Processes which are Stationary in the Strict Sense.- Generalized Random Processes.- Brownian Motion.- Stochastic Integral and Itô Formula.- Stochastic Differential Equations.- A Problem in Homogenization.- Gibbs Random Fields.
